Remark 1.14
For any Dirichlet character
, let
be the entire
-function defined by
the Dirichlet series
The standard
interpolation property of
is that for
any primitive
Dirichlet character
of conductor
(for any
), we
have
|
(1.5.1) |
where
is the Gauss sum:
Note, in particular, that
if and only
if
.
Remark 1.18
Mazur, Tate, and Teitelbaum also define an analogue of
for
primes of bad multiplicative reduction and make
a conjecture. A prime
is
supersingular
for
if
; it is a theorem
of Elkies [
Elk87] that for any elliptic curve
there
are infinitely many supersingular primes
.
Perrin-Riou, Pollack, Greenberg
and others have studied
at good
supersingular primes. More works needs to be done on
finding a definition of
when
is a prime
of bad additive reduction for
.